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Macaulay2Doc :: nullhomotopy

nullhomotopy -- make a null homotopy

Description

nullhomotopy f -- produce a nullhomotopy for a map f of chain complexes.

Whether f is null homotopic is not checked.

Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.

i1 : A = ZZ/101[x,y];
i2 : M = cokernel random(A^3, A^{-2,-2})

o2 = cokernel | 16x2-15xy-26y2  -33x2-24xy+44y2 |
              | -15x2+40xy+26y2 -18x2+20xy+8y2  |
              | -3x2+9xy+31y2   -25x2+18xy+12y2 |

                            3
o2 : A-module, quotient of A
i3 : R = cokernel matrix {{x^3,y^4}}

o3 = cokernel | x3 y4 |

                            1
o3 : A-module, quotient of A
i4 : N = prune (M**R)

o4 = cokernel | 34x2+21y2    -7x2+5xy-29y2 x3 x2y+37xy2+8y3 33xy2-3y3   y4 0  0  |
              | x2-41xy-46y2 -44xy-48y2    0  41xy2         -48xy2+24y3 0  y4 0  |
              | 41xy-13y2    x2-6xy+21y2   0  -5y3          xy2+32y3    0  0  y4 |

                            3
o4 : A-module, quotient of A
i5 : C = resolution N

      3      8      5
o5 = A  <-- A  <-- A  <-- 0
                           
     0      1      2      3

o5 : ChainComplex
i6 : d = C.dd

          3                                                                            8
o6 = 0 : A  <------------------------------------------------------------------------ A  : 1
               | 34x2+21y2    -7x2+5xy-29y2 x3 x2y+37xy2+8y3 33xy2-3y3   y4 0  0  |
               | x2-41xy-46y2 -44xy-48y2    0  41xy2         -48xy2+24y3 0  y4 0  |
               | 41xy-13y2    x2-6xy+21y2   0  -5y3          xy2+32y3    0  0  y4 |

          8                                                                           5
     1 : A  <----------------------------------------------------------------------- A  : 2
               {2} | 8xy2          -48xy2-13y3    -8y3       -17y3     -47y3     |
               {2} | -31xy2-43y3   -12y3          31y3       12y3      -43y3     |
               {3} | -9xy-6y2      21xy+30y2      9y2        -11y2     -17y2     |
               {3} | 9x2-21xy-24y2 -21x2-45xy+7y2 -9xy+27y2  11xy+45y2 17xy-44y2 |
               {3} | 31x2-4xy+15y2 -44xy+10y2     -31xy+47y2 -12xy-4y2 43xy-27y2 |
               {4} | 0             0              x-18y      30y       11y       |
               {4} | 0             0              -8y        x-9y      -43y      |
               {4} | 0             0              -3y        -19y      x+27y     |

          5
     2 : A  <----- 0 : 3
               0

o6 : ChainComplexMap
i7 : s = nullhomotopy (x^3 * id_C)

          8                            3
o7 = 1 : A  <------------------------ A  : 0
               {2} | 0 x+41y 44y  |
               {2} | 0 -41y  x+6y |
               {3} | 1 -34   7    |
               {3} | 0 36    -45  |
               {3} | 0 5     29   |
               {4} | 0 0     0    |
               {4} | 0 0     0    |
               {4} | 0 0     0    |

          5                                                                               8
     2 : A  <--------------------------------------------------------------------------- A  : 1
               {5} | -36 -27 0 8y      -13x-4y  xy-4y2       46xy+10y2   -47xy-8y2   |
               {5} | 44  44  0 24x-48y -20x-18y -41y2        xy-28y2     48xy+45y2   |
               {5} | 0   0   0 0       0        x2+18xy-50y2 -30xy-9y2   -11xy+21y2  |
               {5} | 0   0   0 0       0        8xy+42y2     x2+9xy-49y2 43xy+47y2   |
               {5} | 0   0   0 0       0        3xy+24y2     19xy-28y2   x2-27xy-2y2 |

                   5
     3 : 0 <----- A  : 2
              0

o7 : ChainComplexMap
i8 : s*d + d*s

          3                    3
o8 = 0 : A  <---------------- A  : 0
               | x3 0  0  |
               | 0  x3 0  |
               | 0  0  x3 |

          8                                       8
     1 : A  <----------------------------------- A  : 1
               {2} | x3 0  0  0  0  0  0  0  |
               {2} | 0  x3 0  0  0  0  0  0  |
               {3} | 0  0  x3 0  0  0  0  0  |
               {3} | 0  0  0  x3 0  0  0  0  |
               {3} | 0  0  0  0  x3 0  0  0  |
               {4} | 0  0  0  0  0  x3 0  0  |
               {4} | 0  0  0  0  0  0  x3 0  |
               {4} | 0  0  0  0  0  0  0  x3 |

          5                              5
     2 : A  <-------------------------- A  : 2
               {5} | x3 0  0  0  0  |
               {5} | 0  x3 0  0  0  |
               {5} | 0  0  x3 0  0  |
               {5} | 0  0  0  x3 0  |
               {5} | 0  0  0  0  x3 |

     3 : 0 <----- 0 : 3
              0

o8 : ChainComplexMap
i9 : s^2

          5         3
o9 = 2 : A  <----- A  : 0
               0

                   8
     3 : 0 <----- A  : 1
              0

o9 : ChainComplexMap

Ways to use nullhomotopy :